1. Cops and Robbers1 Conjectures on Cops and Robbers Anthony Bonato Ryerson University Joint Mathematics Meetings AMS Special Session

2. Cops and Robbers 2 C C C R

3. 3 C C C R

4. 4 C C C R cop number c(G) ≤ 3

5. Cops and Robbers played on a reflexive undirected graph G two players Cops C and robber R play at alternate time-steps (cops first) with perfect information players move to vertices along edges; may move to neighbors or pass cops try to capture (i.e. land on) the robber, while robber tries to evade capture minimum number of cops needed to capture the robber is the cop number c(G) –well-defined as c(G) ≤ |V(G)| Cops and Robbers5

6. Conjectures conjectures and problems on Cops and Robbers coming from 5 different directions, touch on various aspects of graph theory: –structural, algorithmic, probabilistic, topological… Cops and Robbers6

7. 1. How big can the cop number be? c(n) = maximum cop number of a connected graph of order n Meyniel Conjecture: c(n) = O(n 1/2 ). Cops and Robbers7

8. 8

9. 9 Henri Meyniel, courtesy Geňa Hahn

10. State-of-the-art (Lu, Peng, 13) proved that –independently proved by (Frieze, Krivelevich, Loh, 11) and (Scott, Sudakov,11) (Bollobás, Kun, Leader,13): if p = p(n) ≥ 2.1log n/ n, then c(G(n,p)) ≤ 160000n 1/2 log n (Prałat,Wormald,14+): proved Meyniel’s conjecture for all p = p(n) Cops and Robbers 10

11. Graph classes (Andreae,86): H-minor free graphs have cop number bounded by a constant. (Joret et al,10): H-free class graphs have bounded cop number iff each component of H is a tree with at most 3 leaves. (Lu,Peng,13): Meyniel’s conjecture holds for diameter 2 graphs, bipartite diameter 3 graphs. Cops and Robbers11

12. Questions Soft Meyniel’s conjecture: for some ε > 0, c(n) = O(n 1-ε ). Meyniel’s conjecture in other graphs classes? –bipartite graphs –diameter 3 –claw-free Cops and Robbers12

13. Cops and Robbers13 How close to n 1/2 ? consider a finite projective plane P –two lines meet in a unique point –two points determine a unique line –exist 4 points, no line contains more than two of them q 2 +q+1 points; each line (point) contains (is incident with) q+1 points (lines) incidence graph (IG) of P: –bipartite graph G(P) with red nodes the points of P and blue nodes the lines of P –a point is joined to a line if it is on that line

14. Example Cops and Robbers14 Fano plane Heawood graph

15. Meyniel extremal families a family of connected graphs (G n : n ≥ 1) is Meyniel extremal if there is a constant d > 0, such that for all n ≥ 1, c(G n ) ≥ dn 1/2 IG of projective planes: girth 6, (q+1)-regular, so have cop number ≥ q+1 –order 2(q 2 +q+1) –Meyniel extremal (must fill in non-prime orders) all other examples of Meyniel extremal families come from combinatorial designs (B,Burgess,2013) Cops and Robbers15

16. Minimum orders M k = minimum order of a k-cop-win graph M 1 = 1, M 2 = 4 M 3 = 10 (Baird, B,12) –see also (Beveridge et al, 14+) M 4 = ? Cops and Robbers16

17. Conjectures on m k, M k Conjecture: M k monotone increasing. m k = minimum order of a connected G such that c(G) ≥ k (Baird, B, 12) m k = Ω(k 2 ) is equivalent to Meyniel’s conjecture. Conjecture: m k = M k for all k ≥ 4. Cops and Robbers17

18. 2. Complexity (Berrarducci, Intrigila, 93), (Hahn,MacGillivray, 06), (B,Chiniforooshan, 09): “c(G) ≤ s?” s fixed: in P; running time O(n 2s+3 ), n = |V(G)| (Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08): if s not fixed, then computing the cop number is NP-hard Cops and Robbers18

19. Questions Goldstein, Reingold Conjecture: if s is not fixed, then computing the cop number is EXPTIME- complete. –same complexity as say, generalized chess settled by (Kinnersley,14+) Conjecture: if s is not fixed, then computing the cop number is not in NP. Cops and Robbers19

20. 3. Genus (Aigner, Fromme, 84) planar graphs (genus 0) have cop number ≤ 3. (Clarke, 02) outerplanar graphs have cop number ≤ 2. Cops and Robbers20

21. Questions characterize planar (outer-planar) graphs with cop number 1,2, and 3 (1 and 2) is the dodecahedron the unique smallest order planar 3-cop-win graph? Cops and Robbers21

22. Higher genus Schroeder’s Conjecture: If G has genus k, then c(G) ≤ k +3. true for k = 0 (Schroeder, 01): true for k = 1 (toroidal graphs) (Quilliot,85): c(G) ≤ 2k +3. (Schroeder,01): c(G) ≤ floor(3k/2) +3. Cops and Robbers22

23. 5. Variants Good guys vs bad guys games in graphs 23 slowmediumfasthelicopter slowtraps, tandem-win, Lazy Cops and Robbers mediumrobot vacuumCops and Robbersedge searching, Cops and Fast Robber eternal security fastcleaningdistance k Cops and Robbers Cops and Robbers on disjoint edge sets The Angel and Devil helicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil, Firefighter Hex bad good

24. Cops and Robbers24 Distance k Cops and Robber (B,Chiniforooshan,09) cops can “shoot” robber at some specified distance k play as in classical game, but capture includes case when robber is distance k from the cops –k = 0 is the classical game C R k = 1

25. Cops and Robbers25 Distance k cop number: c k (G) c k (G) = minimum number of cops needed to capture robber at distance at most k G connected implies c k (G) ≤ diam(G) – 1 for all k ≥ 1, c k (G) ≤ c k-1 (G)

26. When does one cop suffice? (RJN, Winkler, 83), (Quilliot, 78) cop-win graphs ↔ cop-win orderings provide a structural/ordering characterization of cop-win graphs for: –directed graphs –distance k Cops and Robbers –invisible robber; cops can use traps or alarms/photo radar (Clarke et al,00,01,06…) –infinite graphs (Bonato, Hahn, Tardif, 10) Cops and Robbers26

27. Lazy Cops and Robbers Cops and Robbers27

28. Questions on lazy cops Cops and Robbers28

29. 29Cops and Robbers Firefighting

30. A strategy (MacGillivray, Wang, 03): If fire breaks out at (r,c), 1≤r≤c≤n/2, save vertices in following order: (r + 1, c), (r + 1, c + 1), (r + 2, c - 1), (r + 2, c + 2), (r + 3, c -2),(r + 3, c - 3),..., (r + c, 1), (r + c, 2c), (r + c, 2c + 1),..., (r + c, n) –strategy saves n(n-r)-(c-1)(n-c) vertices –strategy is optimal assuming fire breaks out in columns (rows) 1,2, n-1, n Cops and Robbers30

31. ¼ -grid conjecture 31Cops and Robbers

32. Infinite hexagonal grid Conjecture: one firefighter cannot contain a fire in an infinite hexagonal grid. Cops and Robbers32

33. Cops and Robbers33

34. 34 A. Bonato, R.J. Nowakowski, Sketchy Tweets: Ten Minute Conjectures in Graph Theory, The Mathematical Intelligencer 34 (2012) 8-15. 1..